Because \(G\)  is a \(\ell \) -group variety, \(G\) -torsors over \(\eta \)  are \(\ell \) -varieties by [1]}, whose isomorphism classes are classified by \(H^1(\ell ,G)\) . It follows from the triviality of \(H^1(\ell ,G)\)  that \(\operatorname{\underline{Hom}}^\otimes (\omega ^G,\eta )(\ell )\cong G(\ell )\) , hence \(\operatorname{\underline{Hom}}^\otimes (\omega ^G,\eta )(\ell )\ne \emptyset \) . [2]} then implies the second assertion.
